Search CPSC 386 Artificial Intelligence Ellen Walker Hiram College

Problem Solving Agent Goal-based agent (next slide) Must find sequence of actions that will satisfy goal No knowledge of the problem other than definition –No rules –No prior experience

Model-based, Goal Based

Defining the Problem World state –All relevant information about the world (even if it cannot be perceived) –E.g. “vacuum at left, left is dirty, right is clean” Initial state –World state in which the agent starts Goal state –World state in which the problem is solved (can be multiple) Actions –What the agent can do –Table: (current state, action) -> new state Solution –A recommended action sequence that will lead from the current state to a goal state

Searching for a Solution If the initial state is a goal state, return an empty sequence of actions Repeat (until a sequence is found) [SEARCH] –Choose an action sequence –If the sequence leads to a goal, save it. Repeat (until the sequence is empty) [EXECUTE] –Remove the first step from the sequence and execute it.

Environmental requirements Static –so the plan will still work later Fully Observable –so agent knows the initial state Discrete –so agent can enumerate choices Deterministic –so agent can plan a sequence of actions, believing each will lead to a specific intermediate state

State Space Formulation State space –Graph of states connected by actions –Usually too big to be explicitly represented, we create a Successor Function instead Successor (state) -> ((action state) (action state)…) Goal test –Is this state a goal state? Path cost –(we assume) sum of costs of actions in a path Solution –Path from initial state to a goal state –Optimal: least-cost path from initial state to a goal state

Example: Water Pouring Problem –We have two buckets; one holds 4 gallons, and one holds 3 gallons. There are no markings on the buckets. –How can we get exactly 2 gallons into one bucket?

Problem Formulation Initial state –Two empty buckets (0 0) –Goal state (x 2) or (2 x) Actions & successor function –Fill a bucket (x y) -> (3 y) or (x y) -> (x 4) –Empty a bucket (x y) -> (0 y) or (x y) -> (y 0) –Pour from one bucket to another (stop when full) (x y) -> (0 x+y) or (x+y-4, 4) (x y) -> (x+y 0) or (3, x+y-3) Path cost: 1 unit per step

Partial State Space for Water Pouring

Graph Node Formulation State (world state at this node) Parent-node (previous node on path) Action (action taken to generate this node) Path cost, g(n) (from initial state to this node) Depth (number of steps taken so far)

Generic Searching Create a node for the initial state, and put it into the set of unexpanded nodes While the problem is not solved… –Pick an unexpanded node [according to strategy] –Stop (and report the path) if it is a goal state –Expand it (create nodes for every possible action that can be applied to that state)

Terminology Expanded nodes (visited) –Nodes that we have already created child nodes for Save these to avoid repetition Fringe nodes (generated but unvisited) –Nodes that exist but we have not yet created child nodes for them.

Uninformed Search Strategies Random (if you let it run long enough) Depth-first –Choose the most recently generated fringe node –Save fringe nodes on a stack Breadth-first –Choose the least recently generated fringe node –Save fringe nodes on a queue Uniform cost –Choose the least cost fringe node –Save fringe nodes on a priority queue, based on path cost

Maze Example Startabcd efgh ijkl mnoGoal

Paths with Costs A D H B C J G E I F

Variations on DFS Backtracking search –Instead of generating all children at once and putting them on the stack, generate them one at a time –Obvious recursive implementation –Minimal space requirements, since only the current path is stored Depth-limited search –Recursive implementation with a depth limit –If limit has been reached, return immediately –Distinguish between “cutoff” and “failure”

Issues with DFS and BFS DFS can waste a very long time on the wrong path! BFS requires much more space to hold the fringe. –DFS stores approximately 1 path –BFS stores all “current” nodes on all paths simultaneously BFS can expand nodes beyond the solution

Iterative Deepening Search Do a depth-limited search at 1, then at 2, etc. Properties –Finds shortest path (like BFS) –Cost of additional searches isn’t much, in the big-O sense. (Because the number of nodes expanded at a level is comparable to all nodes expanded at earlier levels) –Doesn’t expand nodes beyond solution (which BFS does) –Cost is O(b d ) while BFS is O(b d+1 ) IDS is the preferred uninformed search method when there is a large search space and the depth of the solution is not known.

Bidirectional Search Two searches at once –Initial state toward goal state –Goal state toward initial state Before expanding a node, see if it’s at the fringe of the other tree. –If so, you have found a complete path 2 trees of depth d/2 have much fewer nodes than 1 tree of depth d Space cost is high, though - one tree must be kept in memory for testing against.

Evaluating Search Strategies Completeness –Will a solution always be found if one exists? Optimality –Will the optimal (least cost) solution be found? Time Complexity –How long does it take to find the solution? –Often represented by # nodes expanded Space Complexity –How much memory is needed to perform the search? –Represented by max # nodes stored at once