Warm-up We’ll often have a warm-up exercise for the 10 minutes before class starts. Here’s the first one… Write the pseudo code for breadth first search and depth first search Iterative version, not recursive class TreeNode TreeNode[] children() boolean isGoal() BFS(TreeNode start)… DFS(TreeNode start)…

Announcements Upcoming due dates Account forms Step-by-step videos F 5 pm Proj 0 M 11:59 pm HW 1 Be careful of edX timezone (UTC)! Account forms Pick-up after lecture (if you like) Step-by-step videos Found in edX Courseware (linked on edX Syllabus)

CS 188: Artificial Intelligence Search Please retain proper attribution and the reference to ai.berkeley.edu. Thanks! Instructors: Stuart Russell and Pat Virtue

Today Agents that Plan Ahead Search Problems Uninformed Search Methods Depth-First Search Breadth-First Search Uniform-Cost Search General theme in the class: A goal we have in mind. A mathematical abstraction to formalize this Algorithms that operate on these abstractions

Reflex Agents Reflex agents: Choose action based on current percept (and maybe memory) May have memory or a model of the world’s current state Do not consider the future consequences of their actions Consider how the world IS Examples: blinking your eye (not using your entire thinking capabilities), vacuum cleaner moving towards nearest dirt

Agents that Plan Ahead Planning agents: Spectrum of deliberativeness: Decisions based on predicted consequences of actions Must have a transition model: how the world evolves in response to actions Must formulate a goal Consider how the world WOULD BE Spectrum of deliberativeness: Generate complete, optimal plan offline, then execute Generate a simple, greedy plan, start executing, replan when something goes wrong

Demo Mastermind

Demo Replanning

Search Problems

Search Problems A search problem consists of: A state space For each state, a set Actions(s) of allowable actions A transition model Result(s,a) A step cost function c(s,a,s’) A start state and a goal test A solution is a sequence of actions (a plan) which transforms the start state to a goal state {N, E} 1 N E Often comes back on exams Goal test – sometimes more than one state that satisfies having achieved the goal, for example, “eat all the dots” Abstraction

Search Problems Are Models

Example: Travelling in Romania State space: Cities Actions: Go to adjacent city Transition model Result(A, Go(B)) = B Step cost Distance along road link Start state: Arad Goal test: Is state == Bucharest? Solution? Infinitely many solutions! Some better than others

What’s in a State Space? Problem: Pathing Problem: Eat-All-Dots The real world state includes every last detail of the environment Wrong example for “eat-all-dots”: (x, y, dot count) A search state abstracts away details not needed to solve the problem Problem: Pathing State representation: (x,y) location Actions: NSEW Transition model: update location Goal test: is (x,y)=END Problem: Eat-All-Dots State representation: {(x,y), dot booleans} Actions: NSEW Transition model: update location and possibly a dot boolean Goal test: dots all false MN states MN2MN states

Quiz: Safe Passage Problem: eat all dots while keeping the ghosts perma-scared What does the state representation have to specify? (agent position, dot booleans, power pellet booleans, remaining scared time)

State Space Graphs and Search Trees

State Space Graphs State space graph: A mathematical representation of a search problem Nodes are (abstracted) world configurations Arcs represent transitions resulting from actions The goal test is a set of goal nodes (maybe only one) In a state space graph, each state occurs only once! We can rarely build this full graph in memory (it’s too big), but it’s a useful idea

More Examples Typical finite graph explicitly defined (theoretical CS)

More Examples Typical exponential (or infinite) graph implicitly defined (AI)

Search Trees A search tree: This is now / start Possible futures Different plans that achieve the same state, will be different nodes in the tree. A search tree: A “what if” tree of plans and their outcomes The start state is the root node Children correspond to possible action outcomes Nodes show states, but correspond to PLANS that achieve those states For most problems, we can never actually build the whole tree

Quiz: State Space Graphs vs. Search Trees Consider this 4-state graph: How big is its search tree (from S)? S a b G a S G b ∞ Important: Lots of repeated structure in the search tree!

Tree Search

Search Example: Romania

Searching with a Search Tree Expand out potential plans (tree nodes) Maintain a frontier of partial plans under consideration Try to expand as few tree nodes as possible

General Tree Search Important ideas: function TREE-SEARCH(problem) returns a solution, or failure initialize the frontier using the initial state of problem loop do if the frontier is empty then return failure choose a leaf node and remove it from the frontier if the node contains a goal state then return the corresponding solution expand the chosen node, adding the resulting nodes to the frontier Important ideas: Frontier Expansion Exploration strategy Main question: which frontier nodes to explore?

A Note on Implementation Nodes have state, parent, action, path-cost A child of node by action a has state = result(node.state,a) parent = node action = a path-cost = node.path-cost + step-cost(node.state, a, self.state) Extract solution by tracing back parent pointers, collecting actions

Depth-First Search

Depth-First Search Strategy: expand a deepest node first Implementation: Frontier is a LIFO stack S G d b p q c e h a f r a b c e d f h p r q S a b d p c e h f r q G

Search Algorithm Properties

Search Algorithm Properties Complete: Guaranteed to find a solution if one exists? Optimal: Guaranteed to find the least cost path? Time complexity? Space complexity? Cartoon of search tree: b is the branching factor m is the maximum depth solutions at various depths Number of nodes in entire tree? 1 + b + b2 + …. bm = O(bm) 1 node b … b nodes b2 nodes m tiers bm nodes

Depth-First Search (DFS) Properties What nodes does DFS expand? Some left prefix of the tree. Could process the whole tree! If m is finite, takes time O(bm) How much space does the frontier take? Only has siblings on path to root, so O(bm) Is it complete? m could be infinite, so only if we prevent cycles (more later) Is it optimal? No, it finds the “leftmost” solution, regardless of depth or cost … b 1 node b nodes b2 nodes bm nodes m tiers

Breadth-First Search

Breadth-First Search Strategy: expand a shallowest node first Implementation: Frontier is a FIFO queue S G d b p q c e h a f r Search Tiers S a b d p c e h f r q G

Breadth-First Search (BFS) Properties What nodes does BFS expand? Processes all nodes above shallowest solution Let depth of shallowest solution be s Search takes time O(bs) How much space does the frontier take? Has roughly the last tier, so O(bs) Is it complete? s must be finite if a solution exists, so yes! Is it optimal? Only if costs are all 1 (more on costs later) 1 node b … b nodes s tiers b2 nodes bs nodes bm nodes

Quiz: DFS vs BFS

Quiz: DFS vs BFS When will BFS outperform DFS? When will DFS outperform BFS? [Demo: dfs/bfs maze water (L2D6)]

Video of Demo Maze Water DFS/BFS (part 1)

Video of Demo Maze Water DFS/BFS (part 2)

Iterative Deepening Idea: get DFS’s space advantage with BFS’s time / shallow-solution advantages Run a DFS with depth limit 1. If no solution… Run a DFS with depth limit 2. If no solution… Run a DFS with depth limit 3. ….. Isn’t that wastefully redundant? Generally most work happens in the lowest level searched, so not so bad! b …

Finding a Least-Cost Path START GOAL d b p q c e h a f r 2 9 8 1 3 4 15 BFS finds the shortest path in terms of number of actions. It does not find the least-cost path. We will now cover a similar algorithm which does find the least-cost path.

Uniform Cost Search

Uniform Cost Search S G 2 Strategy: expand a cheapest node first: b p q c e h a f r 2 Strategy: expand a cheapest node first: Frontier is a priority queue (priority: cumulative cost) 1 8 2 2 3 9 2 8 1 1 15 S a b d p c e h f r q G 9 1 3 4 5 17 11 16 11 Cost contours 6 13 7 8 11 10

Uniform Cost Search (UCS) Properties What nodes does UCS expand? Processes all nodes with cost less than cheapest solution! If that solution costs C* and arcs cost at least  , then the “effective depth” is roughly C*/ Takes time O(bC*/) (exponential in effective depth) How much space does the frontier take? Has roughly the last tier, so O(bC*/) Is it complete? Assuming best solution has a finite cost and minimum arc cost is positive, yes! Is it optimal? Yes! (Proof next lecture via A*) b c  1 … c  2 C*/ “tiers” c  3

Uniform Cost Issues Remember: UCS explores increasing cost contours … c  3 c  2 c  1 Remember: UCS explores increasing cost contours The good: UCS is complete and optimal! The bad: Explores options in every “direction” No information about goal location We’ll fix that soon! Start Goal

Video of Demo Empty UCS

Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 1) Let students guess which one of the three it is. (This is BFS.)

Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 2) Let students guess which one of the three it is. (This is UCS.)

Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 3) Let students guess which one of the three it is. (This is DFS.)

Tree Search vs Graph Search

Tree Search: Extra Work! Failure to detect repeated states can cause exponentially more work. State Graph Search Tree O(2m) O(m)

Tree Search: Extra Work! Failure to detect repeated states can cause exponentially more work. State Graph Search Tree O(4m) O(2m2) m=20: 800 states 1,099,511,627,776 tree nodes

General Tree Search function TREE-SEARCH(problem) returns a solution, or failure initialize the frontier using the initial state of problem loop do if the frontier is empty then return failure choose a leaf node and remove it from the frontier if the node contains a goal state then return the corresponding solution expand the chosen node, adding each child to the frontier

General Graph Search function GRAPH-SEARCH(problem) returns a solution, or failure initialize the frontier using the initial state of problem initialize the explored set to be empty loop do if the frontier is empty then return failure choose a leaf node and remove it from the frontier if the node contains a goal state then return the corresponding solution add the node to the explored set expand the chosen node, adding each child to the frontier but only if the child is not already in the frontier or explored set Theorem: each state appears at most once in the search tree constructed

Tree Search vs Graph Search Avoids infinite loops: m is finite in a finite state space Eliminates exponentially many redundant paths Requires memory proportional to its runtime!

Search Gone Wrong?