Homework One
Search
1 Syllabus Acknowledgement
Read the entire syllabus and write me a short note acknowledging that you did this. Specifically acknowledge the schedule of deliverables and exams, the readings, and the collaboration policy. Let me know how prepared you feel to take the course. Include this document as a PDF in your homework submission.
2 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.
This project includes an autograder for you to grade your answers locally. This can be run with the command:
python autograder.pyThe 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 in: search dot zip.
Files to Edit and Submit: You will fill in portions of search.py and searchAgents.py during the assignment. Once you have completed the assignment, you will submit these files to the submit server. Please submit a single (uncompressed) .tar containing these two files and your syllabus acknowledgement PDF. Include a comment at the start of searchAgents.py indicating what help, if any, you got on this assignment. If you didn’t get any help, indicate this. Failure to include this comment will result in losing 10 points on the assignment grade.
Evaluation: Evaluation of your code’s correctness will be performed by the autograder. If you think your code is correct and the autograder is in error, bring this to my attention before the submission deadline. Point values are relative within each assignment: all assignments are scaled to 100 when calculating grades. The server will allow you to submit multiple times for credit, but please run the autograder locally first.
Academic Dishonesty: This assignment is individual effort. Please review the collaboration policy for the course and adhere to it.
| 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 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. |
| You can ignore other supporting files. |
3 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.pyPacman 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 GoWestAgentBut, things get ugly for this agent when turning is required:
python pacman.py --layout tinyMaze --pacman GoWestAgentIf 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 -h4 Q1 (3 pts): Depth First Search
(3 pts) 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=tinyMazeSearchThe 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).
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 SearchAgentThe 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.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q15 Q2 (3 pts): 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 .5Does 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.pyGrading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q26 Q3 (3 pts): 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 StayWestSearchAgentNote: 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).
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q37 Q4 (3 pts): 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=manhattanHeuristicYou 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?
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q48 Q5 (3 pts): 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=CornersProblemTo 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).
An instance of the CornersProblem class represents an entire search problem, not a particular state. Particular states are returned by the functions you write, and your functions return a data structure of your choosing (e.g. tuple, set, etc.) that represents a state.
Furthermore, while a program is running, remember that many states simultaneously exist, all on the queue of the search algorithm, and they should be independent of each other. In other words, you should not have only one state for the entire CornersProblem object; your class should be able to generate many different states to provide to the search algorithm.
Hint 1: 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.
Hint 2: When coding up getSuccessors, make sure to add children to your successors list with a cost of 1.
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.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q5It’s possible to code a “heuristic” that finds actual path cost using DFS, BFS, or UCS, and doesn’t count those nodes as expansions for the autograder. Please don’t do this. You won’t get credit, even if you initially pass the autograder.
9 Q6 (3 pts): 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 0.5Note: AStarCornersAgent is a shortcut for
-p SearchAgent -a fn=aStarSearch,prob=CornersProblem,heuristic=cornersHeuristicAdmissibility 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 usually 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 | Grade |
|---|---|
| more than 2000 | 0/3 |
| at most 2000 | 1/3 |
| at most 1600 | 2/3 |
| at most 1200 | 3/3 |
Remember: If your heuristic is inconsistent, you will receive no credit, so be careful!
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q610 Q7 (4 pts): 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 AStarFoodSearchAgentNote: AStarFoodSearchAgent is a shortcut for
-p SearchAgent -a fn=astar,prob=FoodSearchProblem,heuristic=foodHeuristicNote: 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 consistent heuristic for the FoodSearchProblem. Try your agent on the trickySearch board:
python pacman.py -l trickySearch -p AStarFoodSearchAgentOur UCS agent finds the optimal solution after expanding 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 | 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) |
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.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q711 Q8 (3 pts): 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 .5Hint: 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.
Grading: Please run the below command to see if your implementation passes all the autograder test cases.
python autograder.py -q q8