P1: Search Algorithms Solution

In this assignment, you will implement search algorithms to help Pacman find paths in a maze. Note: We will use the Pacman framework developed at Berkeley. This framework is used worldwide to teach AI, therefore it is very important that you DO NOT publish your solutions online.

Written Assignment (12 points)

As usual please go to the PDF in the written folder for the written instructions.

Question 1 (3 points)

Question 2 (3 points)

Question 3 (3 points)

Question 4 (3 points)

Pacman (25 points + 1 bonus)


























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.

Welcome to the first of 4 Pacman problem sets this semester. 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. For those of you who aren't familiar with, he is a character from an old video game that runs around a maze trying to capture all the dots without getting eaten by the ghosts. Pacman can also eat the special large dots to powerup into ghost eating mode. Throughout the semester we will code up more and more sophisticated Pacman agents based on the algorithms we are learning about in class.

While these problem sets follow the Berkely material closely we wil sometimes provide additional hints in our instructions and will sometimes NOT do part of the Berkeley assignment. Therefore please always follow the instructions given in these README files. Also even if you are working in partners BOTH of you must submit your code to Gradescope before the deadline!

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.

The rest of the files are simply supporting files for graphics and scaffolding for various game play and grading operations.

Introduction

Welcome to Pacman

After downloading the code 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    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 .


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 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 Stack , Queue and 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 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 but you may find it very helpful).

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 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 depth-first search is doing wrong.


REMEMBER TO RUN python autograder.py -q q1 to make sure you pass all the grading tests for q1!

Question 2 (3 points): Breadth First Search


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 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 eight-puzzle search problem without any changes.


python eightpuzzle.py


As always remember to run python autograder.py -q q2 to make sure you pass all the grading tests for q2!

Question 3 (3 points): 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 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 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).


As always remember to run python autograder.py -q q3 to make sure you pass all the grading tests for q3!

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?


As always remember to run python autograder.py -q q4 to make sure you pass all the grading tests for q4!

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.


As always remember to run python autograder.py -q q5 to make sure you pass all the grading tests for q4!

Question 6 (3 points): 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 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: 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 shown below.

Remember: If your heuristic is inconsistent, you will receive no credit, so be careful!


Number of nodes expanded

Grade

more than 2000

0/3

at most 2000
1/3



at most
1600
2/3



at most
1200
3/3





As always remember to run python autograder.py -q q6 to make sure you pass all the grading tests for q6!

Question 7 (4 points + 1 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 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 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 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 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 FoodSearchProblem . Try your agent on the

with a consistent heuristic for 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 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.

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.


Number of nodes expanded

Grade

more than 15000

1/4

at most 15000

2/4

at most 12000

3/4

at most 9000

4/4 (full credit; medium)


at most 7000

5/4 (optional extra credit; hard)


As always remember to run python autograder.py -q q7 to make sure you pass all the grading tests for q7!

Question 8 (3 points): 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 greedily 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.


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. Make sure you understand why and try to come up with a small example where repeatedly going to the closest dot does not result in finding the shortest path for eating all the dots.


As always remember to run python autograder.py -q q8 to make sure you pass all the grading tests for q8!


To test that every is done (and what our grading script will do) run python autograder.py .


Important Note:

The autograder is king so make sure that your implementation follows the spec asked for by the instructions. If you are running into errors maybe you are avoiding pushing valid paths to the frontier because you are making an optimization that is valid in the BFS case but not in the A* case. If the spec did not ask for this optimization it may cause the autograder to fail. This is another reason why we suggest using a general search! There will be less things to debug and less areas to do a weird quirk if you are trying to be generic! Also you should not have to modify the stack, queue, or priority queue data structures at all to accomplish this homework!

Finally you can run the autograder as many times as you'd like before submitting and can submit multiple times as well just make sure the final code passes all of the tests via python autograder.py !


You must submit to Gradescope to get credit for the assignment --- make sure you submit the required files before the deadline!