HW1: Search: Due 2/3/2022
Table of Contents
All those colored walls,
Mazes give Pacman the blues,
So teach him to search.
In this assignment, 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.
As in HW 0, this assignment includes an autograder for you to grade your answers on your machine. This can be run with the command:
See the autograder tutorial in HW 0 for more information about using the autograder.
The code for this assignment 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 as a zip archive.
|Files you'll edit:
|Where all of your search algorithms will reside.
|Where all of your search-based agents will reside.
|Files you might want to look at:
|The main file that runs Pacman games. This file describes a Pacman GameState type, which you use in this assignment.
|The logic behind how the Pacman world works. This file describes several supporting types like AgentState, Agent, Direction, and Grid.
|Useful data structures for implementing search algorithms.
|Supporting files you can ignore:
|Graphics for Pacman
|Support for Pacman graphics
|ASCII graphics for Pacman
|Agents to control ghosts
|Keyboard interfaces to control Pacman
|Code for reading layout files and storing their contents
|Parses autograder test and solution files
|General autograding test classes
|Directory containing the test cases for each question
|HW 1 specific autograding test classes
Files to Edit and Submit: You will fill in portions of
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. Do not submit code that imports other libraries; these will not be available to the autograder on Gradescope.
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 judgements -- will be the final judge of your score. If necessary, we will review and grade assignments individually to ensure that you receive due credit for your work.
Academic Dishonesty: We will be checking your code against other submissions, both within and outside of this year's class. If you copy someone else's code and submit it with minor changes, we will know. These cheat detectors are quite hard to fool, so please don't try. We trust you all to submit your own work only; please don't let us down. If you do, we will pursue the strongest consequences available to us.
Getting Help: You are not alone! If you find yourself stuck on something, contact the course staff for help. Office hours, section, and the discussion forum are there for your support; please use them. If you can't make our office hours, let us know and we will try to work something out for you. We want these assignments to be rewarding and instructional, not frustrating and demoralizing. But, we don't know when or how to help unless you ask.
Ed discussion: Please be careful not to post spoilers. Post anything that reveals answers in private posts. By now, you should have the skills to debug your own code, but if you find it necessary to show us your code, privately post your code. DO NOT POST A PICTURE OF YOUR CODE. That makes it unnecessarily difficult for us to help you.
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:
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 CTRL-c into your terminal.
Soon, your agent will solve not only
tinyMaze, but any maze you want.
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 assignment 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
searchAgents.py, you'll find a fully implemented
which plans out a path through Pacman's world and then executes that
path step-by-step. The search algorithms for formulating a plan are not
implemented -- that's your job.
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 full-fledged 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
PriorityQueue data structures provided to you in
util.py! These data structure implementations have particular properties which are required for compatibility with the autograder.
Hint: Each algorithm is very similar. Algorithms for DFS, BFS, UCS, and A* differ only in the details of how the frontier is managed. 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).
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 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
should have a length of 130 (provided you push successors onto the
frontier 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 depth-first search is doing wrong.
Hint: The autograder can be a bit fiddly for this question, and it will complain if your implementation doesn't do things exactly as it expects. For DFS, add states to the visited when you expand, and be sure to test if a state has been visited before pushing, and to test when you pop.
Implement the breadth-first search (BFS) algorithm in the
breadthFirstSearch function in
Again, write a graph search algorithm that avoids expanding any already
visited states. Test your code the same way you did for depth-first
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
Note: If you've written your search code generically, your code should work equally well for the eight-puzzle search problem without any changes.
Hint: The autograder can also be a bit fiddly for this question, and it will complain if your implementation doesn't do things exactly as it expects. As with DFS, add states to the visited when you expand them, and be sure to test if a state has been visited before pushing, and to test when you pop.
While BFS will find a fewest-actions path to the goal, we might want to find paths that are "best" in other senses. Consider
By changing the cost function, we can encourage Pacman 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 Pacman agent should adjust its behavior in response.
Implement the uniform-cost graph search algorithm in the
uniformCostSearch function in
search.py. We encourage you to look through
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
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
StayWestSearchAgent respectively, due to their exponential cost functions (see
searchAgents.py for details).
Implement A* graph search in the empty function
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
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? (This is for you to ponder; you are not required to submit an expilcit aswer to this question.)
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.
CornersProblem search problem in
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.
Note: Make sure to complete Question 4 before working on Question 6, because Question 6 builds upon your answer for Question 4.
Implement a non-trivial, consistent heuristic for the
python pacman.py -l mediumCorners -p AStarCornersAgent -z 0.5
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: 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:
|Number of nodes expanded
|more than 2000
|at most 2000
|at most 1600
Remember: If your heuristic is inconsistent, you will receive no credit, so be careful!
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 food-clearing problem:
(implemented for you). A solution is defined to be a path that collects
all of the food in the Pacman world. For the present assignment, 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
adversarial search.) If you have written your general search methods
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
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.
searchAgents.py with a consistent heuristic for the
FoodSearchProblem. Try your agent on the
python pacman.py -l trickySearch -p AStarFoodSearchAgent
Our UCS agent finds the optimal solution in about 13 seconds, exploring over 16,000 nodes.
Any non-trivial non-negative 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
|more than 15000
|at most 15000
|at most 12000
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.
This question asks you to verify your understanding of A* by working through a simple example by hand. We will use tinyMaze with the Manhattan distance as your heuristic. You can find a text descriptin of tinyMaze in the layouts folder as "tinyMaze.lay". The goal position is represented with a "." and starting position is represented with a "P". Walls are represented with "%". The representation for states is in (X,Y) format, with the (0,0) origin in the bottom left. Thus, the goal is in position (1,1), and the empty square to the right in position (2,1).
The main thing for you to turn in for this problem is the final status of the queue after the goal is popped off. You should show all states still on the queue, along with their f value. You are not required to show intermediate points in the search, but you are encouraged to do so, as it makes it easier to give partial credit. (This is a tiny problem, so don't be surprised if your answer is pretty short.)
In principle, you could simply run your A* code, print out the queue and then copy that over as your solution. We will have no way of knowing this is what you have done, but you are strongly encouraged to do things in the opposite order: Work out this solution by hand and then compare with what your A* code does. This will deepen your understanding of A* and the experience will make it easier to debug your code.
If you encounter any ties, indicate any assumptions you have made about how ties are broken.
Note: For working this out by hand, your queue will be much smaller if you check the visited list before pushing, i.e., as described in the slides and text. You will get credit for doing it either way though.
Consider the problem of finding the shortest path for an Amazon driver to deliver some packages. Assume the driver is constrained to move on a graph where the vertices have coordinates that correspond to positions in 2D space, e.g., house locations and intersection locations.
a) Suppose there are n vertices in the graph and k packages to be delivered, where each location is to receive at most one package. What is the size of the state space? (Assume that each package can be delivered to one and only one house, so don't worry about counting the number of different ways packages can get misdelivered.)
b) Consider the following heuristic: h = the sum of the Euclidian distances from the current position to each of the unvisited package drop off points. Prove that this is not admissible by counterexample.
c) Consider the following heuristic: h = the sum of the Euclidean distance from the current position to the closest unvisited drop off point, plus the Euclidean distance from there to next closest drop off point to it, and so on. This is much more clever than the previous heuristic because it takes into account the closest next drop off points rather than measuring everything from the current position. It turns out, however, that this sill isn't admissible. Prove this by counterexample.
Example of how this works: Suppose S is the start state, and there are three packages (k=3) that need to be delivered to locations X, Y, and Z. Assume X is the closest location to S. Choosing between Y and Z, assume Z is closest to X. We would have h(s) = d(S,X) + d(X,Z) + d(Z,Y), where d(.,.) is the Euclidean distance. Locations in your graph that are not destinations for packages should not be consisdered when computing h.
If g(x) and h(x) are both consistent is, f(x)=max(g(x),h(x)) necessairily consistent? Disprove this by example or prove it mathematically. Your answer should be concise - not an essay.
To submit your homework, please upload the
following files to HW 1 (code) on Gradescope:
searchAgents.py. Please do not upload
the files in a zip file or a directory as the autograder
will not work if you do so.
Separately, please upload a single pdf for quesitons 8-10 to HW 1 on Gradescope.