Homework 1: Search in Pacman

All those colored walls,
Mazes give Pacman the blues,
So teach him to search.

Introduction

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.

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 (including this description) as a zip archive.

What to submit: You will fill in portions of search.py and searchAgents.py during the assignment. You should submit these two files and a PDF of the proof we request below in a single zip file via Sakai.

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. If necessary, we will review and grade assignments individually to ensure that you receive due credit for your work.

Academic Dishonesty: You may discuss questions with your classmates, but any written code or text you submit must be your own work. You may not copy code or text from your classmates or from the internet.

Getting Help: You are not alone! If you find yourself stuck on something, contact us for help. Office hours and Piazza are there for your support; please use them. If you can't make our office hours, let us know and we will schedule more. We want these assignments to be rewarding and instructional, not frustrating and demoralizing. However, 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 Pacman

python pacman.py

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

python pacman.py --layout tinyMaze --pacman GoWestAgent

python pacman.py -h

Finding a Fixed Food Dot using Search Algorithms

python pacman.py -l tinyMaze -p SearchAgent -a fn=tinyMazeSearch

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 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).

Hint: Each algorithm is very similar. Algorithms for DFS, BFS, UCS, and A* differ only in the details of how the frontier 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).

Hint: Make sure to check out the Stack, Queue and PriorityQueue types provided to you in util.py!

Question 1 (10 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 (textbook section 3.5).

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

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 244 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.

Question 2 (10 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

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 (textbook section 3.2) without any changes.

python eightpuzzle.py

Varying the Cost Function

Question 3 (10 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 (10 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 

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 (10 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

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 (10 points) Implement a non-trivial, consistent heuristic for the CornersProblem in cornersHeuristic. Grading: inconsistent heuristics will get no credit. 3 points for any non-trivial consistent heuristic, 3 points for expanding fewer than 1600 nodes and 3 points for expanding fewer than 1200 nodes. Expand fewer than 800, and you're doing a great job!

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: In class, we focused on tree search, but there are some subtleties in applying A* to graph search. (Read pp. 94-95 of the text carefully.) 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. Morevoer, if UCS and A* ever return paths of different lengths, your heuristic is inconsistent.

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).

Additionally, any heuristic should always be non-negative, and should return a value of 0 at every goal state (technically this is a requirement for admissibility!). We will deduct 3 points for any heuristic that returns negative values, or doesn't behave properly at goal states.

Question 7 (10 points) Prove that consistency implies admissibility and give an example of an admissible heuristic that is not consistent (problem 3.29 from the text).

Eating All The Dots

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 8 (10 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

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.

We will deduct 1 point for any heuristic that returns negative values, or does not return 0 at every goal state.

Beyond Pacman

Question 9 (5 points) Show that an inadmissable heuristic can lead to A* returning a suboptimal example, by giving an example search tree and heuristic where this occurs. Aim for as small a tree as possible, and explain your example clearly.

Question 10 (10 points) Give a detailed example (by drawing the states and labeling the edges with costs) where negative edge costs could lead to the following parodoxical result: a finite state space with an optimal solution that does not reach the goal in finite time. Be sure to explain why your example has this property.

Question 11 (5 points) Suppose you are asked to apply search to a symbol problem. You are told that the solution is a sequence of at most five symbols, but the symbols are drawn from an alphabet which contains thousands of symbols. Formulate this problem as a search problem. If depth-first and breadth-first were your own choices, which would you apply? Explain. How might you change the algorithm?

Question 12 (5 points) Modify iterative deepening search so that it finds the lowest cost, as opposed to shortest, path. What information do you need? Prove that your algorithm finds the optimal solution.

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 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.