Homework 1: Search in Pac-Man

The Pacman AI projects were developed at UC Berkeley, primarily by John DeNero (denero@cs.berkeley.edu) and Dan Klein (klein@cs.berkeley.edu). For more info, see http://inst.eecs.berkeley.edu/~cs188/sp09/pacman.html

All those colored walls,
Mazes give Pac-Man the blues,
So teach him to search.

Introduction

In this project, your Pac-Man 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 Pac-Man scenarios.

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. You can download all the code and supporting files (including this description) as a zip archive.

Files you'll edit:
search.py Where all of your search algorithms will reside.
searchAgents.py Where all of your search-based agents will reside.
Files you might want to look at:
pacman.py The main file that runs Pac-Man games. This file describes a Pac-Man GameState type, which you use in this project.
game.py The logic behind how the Pac-Man 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 Pac-Man
graphicsUtils.py Support for Pac-Man graphics
textDisplay.py ASCII graphics for Pac-Man
ghostAgents.py Agents to control ghosts
keyboardAgents.py Keyboard interfaces to control Pac-Man
layout.py Code for reading layout files and storing their contents

What to submit: You will fill in portions of search.py and searchAgents.py during the assignment.

Evaluation: Your code will be autograded for technical correctness. Please do not change the names of any provided functions or classes within the code, or you will wreak havoc on the autograder. However, the correctness of your implementation -- not the autograder's output -- will be the final judge of your score.

Getting Help: You are not alone! If you find yourself stuck on something, contact the course staff for help. If you can't make our office hours, let us know and we will schedule more. We want these projects to be rewarding and instructional, not frustrating and demoralizing. But, we don't know when or how to help unless you ask. One more piece of advice: if you don't know what a variable does or what kind of values it takes, print it out.

Welcome to Pac-Man

After downloading the code (search.zip), unzipping it and changing to the search directory, you should be able to play a game of Pac-Man by typing the following at the command line: 
python pacman.py
Pac-Man lives in a shiny blue world of twisting corridors and tasty round treats. Navigating this world efficiently will be Pac-Man'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 CTRL-c 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.

Finding a Fixed Food Dot using Search Algorithms

In searchAgents.py, you'll find a fully implemented SearchAgent, which plans out a path through Pac-Man's world and then executes that path step-by-step. The search algorithms for formulating a plan are not implemented -- that's your job. As you work through the following questions, you might need to refer to this glossary of objects in the code. 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. Pac-Man should navigate the maze successfully.

Now it's time to write full-fledged generic search functions to help Pac-Man plan routes! Pseudocode for the search algorithms you'll write can be found in the lecture slides and textbook. 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 types provided to you in util.py! This is required in order for the autograder to function properly.

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 algorithm-specific queuing strategy. (Your implementation need not be of this form to receive full credit).

Question 1 (2 points) Implement the depth-first 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 Pac-Man 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 Pac-Man actually go to all the explored squares on his way to the goal?

Question 2 (2 points) Implement the breadth-first 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 depth-first 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 Pac-Man 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 eight-puzzle search problem (textbook section 3.2) without any changes. 

python eightpuzzle.py

Varying the Cost Function

While BFS will find a fewest-actions 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 Pac-Man to find different paths. For example, we can charge more for dangerous steps in ghost-ridden areas or less for steps in food-rich areas, and a rational Pac-Man agent should adjust its behavior in response.

Question 3 (2 points) Implement the uniform-cost 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).

A* search

Question 4 (3 points) 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?

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.

Question 5 (2 points) 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 Pac-Man 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 Pac-Man 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 (3 points) Implement a non-trivial, 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 non-negative). 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 f-value. Moreover, if UCS and A* ever return paths of different lengths, your heuristic is inconsistent. This stuff is tricky!

Non-Trivial 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 the autograder. You want a heuristic which reduces total compute time, though for this assignment the autograder will only check node counts (aside from enforcing a reasonable time limit).

Grading: 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 the autograder. You want a heuristic which reduces total compute time, though for this assignment the autograder will only check node counts (aside from enforcing a reasonable time limit).

Your heuristic must be a non-trivial non-negative 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 as follows. More than 2000 nodes expanded gives zero credit. At most 2000 gives 1 point. At most 1600 gives 2 points. At most 1200 gives full credit.

Eating All The Dots

Now we'll solve a hard search problem: eating all the Pac-Man food in as few steps as possible. For this, we'll need a new search problem definition which formalizes the food-clearing problem: FoodSearchProblem in searchAgents.py (implemented for you). A solution is defined to be a path that collects all of the food in the Pac-Man 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 Pac-Man. (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 uniform-cost 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 4902 search nodes.

Question 7 (4 points) Fill in foodHeuristic in searchAgents.py with a consistent heuristic for the 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. If your heuristic is admissible, you will receive the following score, depending on how many nodes your heuristic expands.
Fewer nodes than:Points
150004
120008
900012 (medium)
700016 (hard)

If your heuristic is inadmissible, you will receive no credit, so be careful!

Suboptimal Search

Sometimes, even with A* and a good heuristic, finding the optimal path through all the dots is hard. In these cases, we'd still like to find a reasonably good path, quickly. In this section, you'll write an agent that always eats the closest dot. ClosestDotSearchAgent is implemented for you in searchAgents.py, but it's missing a key function that finds a path to the closest dot.

Question 8 (2 points) Implement the function findPathToClosestDot in searchAgents.py. Our agent solves this maze (suboptimally!) in under a second with a path cost of 350: 

python pacman.py -l bigSearch -p ClosestDotSearchAgent -z .5

Hint: The quickest way to complete findPathToClosestDot is to fill in the AnyFoodSearchProblem, which is missing its goal test. Then, solve that problem with an appropriate search function. The solution should be very short!

Your ClosestDotSearchAgent won't always find the shortest possible path through the maze. (If you don't understand why, ask a GSI!) In fact, you can do better if you try.

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 and breadthFirstSearch, which you have to write. You are provided tinyMazeSearch which is a very bad search function that only works correctly on tinyMaze
SearchAgent
SearchAgent is is a class which implements an Agent (an object that interacts with the world) and does its planning through a search function. The SearchAgent 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.