There are eight exercises in this lab:
1: DFS
2: BFS
3: UCS
4: A*
5: CornersProblem
6: CornersProblem heuristic
7: FoodSearchProblem heuristic
8: Sub-optimal search
setup63
to create a git
repository for the lab. If you want to work alone do:
setup63 labs/02 noneIf you want to work with a partner, then one of you needs to run the following while the other one waits until it finishes.
setup63 labs/02 partnerUsernameOnce the script finishes, the other partner should run it on their account.
cd ~/cs63/labs/02 cp -r ~meeden/public/cs63/labs/02/* ./This will copy over the starting point files for this lab.
git add * git commit -m "lab2 start" git push
cd ~/cs63/labs/02 git pull
In this lab, 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 many different Pac-Man scenarios.
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 should look at but NOT edit: | |
util.py |
Useful data structures for implementing search algorithms. |
pacman.py |
The main file that runs Pac-Man games. This file describes a Pac-Man GameState type, which you use in this lab. |
game.py |
The logic behind how the Pac-Man world works. This file describes several supporting types like AgentState, Agent, Direction, and Grid. |
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=tinyMazeSearchThe command above tells the
SearchAgent
to
use tinyMazeSearch
as its search algorithm, which is
implemented in search.py
. This simply follows a fixed
sequence of actions to demonstrate how the code works. 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 depth-first search and breadth-first search algorithms you'll write is shown below.
UninformedSearch(problem) returns a list of actions initialize the frontier using the initial state of the problem #For explored, use Pacman position as the key with a value True initialize a dictionary of states already explored loop if the frontier is empty return an empty list choose a leaf node and remove it from the frontier if the node contains a goal state return list of actions from start state to goal state add the state key to the explored dictionary for each successor of the node state if the key of the successor state is not in explored add node of the successor onto the frontier
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: Algorithms for DFS and BFS differ only in the details of how the fringe is managed. So, concentrate on getting DFS right and then BFS 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
.
Exercise 1 Implement the
depth-first search algorithm in the
depthFirstSearch
function in search.py
. You
should begin by creating a Node
class to use in all of
your search algorithms. Recall that a search node contains:
Although DFS and BFS ignore the costs, you'll need them for later search methods.
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 Pac-Man board will show an overlay of color for 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?
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.
Exercise 2 Implement the
breadth-first search algorithm in the
breadthFirstSearch
function in search.py
.
Use the same algorithm as shown in the above pseudocode. 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 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
Remember to use git to add, commit, and push your updates before going on to the next exercise.
mediumDottedMaze
and mediumScaryMaze
, which you can find
in the layouts
directory. 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.
To properly implement informed search we need to update our search
pseudocode. One key difference is that the frontier will be
implemented as a priority queue. We will choose the node from the
frontier with the lowest path cost. Another important difference is
that we will need to track which states are on the frontier. If we
find a new path to the same state that has a lower path cost, we want
to be sure to focus on that new node. To do this we will add a second
dictionary called considering
. The updated pseudocode is
given below:
InformedSearch(problem) returns a list of actions initialize a PQ frontier using the initial state of the problem #For explored, use Pacman position as the key, value True initialize a dictionary of states already explored #For considering, use Pacman position as the key, value pathCost initialize a dictionary of states being considered loop if the frontier is empty return an empty list choose a leaf node and remove it from the frontier if node state is in explored continue with next iteration of the loop #Ignore it add state key to explored dictionary remove state key from considering dictionary if the node contains a goal state return list of actions from start state to goal state for each successor of the node state calculate the new pathCost if key of successor state not in explored and considering add key of successor state to considering with pathCost push the successor node on the frontier elif key of successor state in considering and new pathCost < old pathCost #Found a better path to a state we were considering update the considering dictionary with the new pathCost push the successor node on the frontier
Exercise 3 Implement the uniform-cost
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).
Remember to use git to add, commit, and push your updates before going on to the next exercise.
Exercise 4 Implement A* search
in the empty function aStarSearch
in search.py
. A* should be implemented using the same
InformedSearch pseudocode given above. One difference between A* and
UCS is that 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. A* uses this
heuristic function to estimate the distance from a state to the goal.
The PQ should sort nodes based on the sum of the cost to get to the
node plus the estimated distance to the goal.
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?
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.
Exercise 5 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 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.
Exercise 6 Implement a heuristic for the CornersProblem
in cornersHeuristic
. Grading: inadmissible heuristics will get no credit. 1 point for any admissible heuristic. 1 point for expanding fewer than 1600 nodes. 1 point for expanding fewer than 1200 nodes. Expand fewer than 800, and you're doing great!
python pacman.py -l mediumCorners -p AStarCornersAgent -z 0.5
Hint: Heuristic functions just return numbers, which, to be admissible, must be lower bounds on the actual shortest path cost to the nearest goal.
Note: AStarCornersAgent
is a shortcut for -p SearchAgent -a fn=aStarSearch,prob=CornersProblem,heuristic=cornersHeuristic
.
Remember to use git to add, commit, and push your updates before going on to the next exercise.
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 lab, 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 a later lab.) 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.
Exercise 7 Fill in foodHeuristic
in searchAgents.py
with a FoodSearchProblem
. Try your agent on the trickySearch
board:
python pacman.py -l trickySearch -p AStarFoodSearchAgentOur UCS agent finds the optimal solution in about 13 seconds, exploring over 16,000 nodes. If your heuristic is admissible, you will receive 1 point. You will receive additional points, depending on how many nodes your heuristic expands:
Nodes expanded | Additional points |
---|---|
12000 > n < 15000 | 1 |
9000 > n < 12000 | 2 |
n < 9000 | 3 |
If your heuristic is inadmissible, you will receive no credit, so be careful! Think through admissibility carefully, as inadmissible heuristics may manage to produce fast searches and even optimal paths. Can you solve mediumSearch
in a short time? If so, we're either very, very impressed, or your heuristic is inadmissible.
Admissibility vs. Consistency? Technically, admissibility isn't enough to guarantee correctness in graph search -- you need the stronger condition of consistency. For a heuristic to be consistent, it must hold that if an action has cost c, then taking that action can only cause a drop in heuristic of at most c. If your heuristic is not only admissible, but also consistent, you will receive 1 additional point for this question.
Almost always, admissible heuristics are also consistent, especially if they are derived from problem relaxations. Therefore it is probably 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. Inconsistency can sometimes be detected by verifying that your returned solutions are non-decreasing in f-value. Moreover, if UCS and A* ever return paths of different lengths, your heuristic is inconsistent.
Remember to use git to add, commit, and push your updates before going on to the next exercise.
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.
Exercise 8 Implement the function findPathToClosestDot
in searchAgents.py
. Our agent solves this maze (sub-optimally!) 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 necessarily find the shortest possible path through the maze. In fact, you can do better if you try.
search.py
and searchAgents.py
.
cd ~/cs63/labs/02 git add search.py searchAgents.py git commit -m "the latest version" git push
Here's a glossary of the key objects in the code base related to search problems, for your reference:
SearchProblem (search.py)
search.py
PositionSearchProblem (searchAgents.py)
CornersProblem (searchAgents.py)
FoodSearchProblem (searchAgents.py)
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.