Search ProblemsSearch Problems

Russell and Norvig: Chapter 3, Sections 3.1 – 3.3

Slides adapted from:robotics.stanford.edu/~latombe/cs121/2003/home.htmby Prof. Jean-Claude Latombe

Why Search?

In many tasks, we know what a solution looks like, but do not have an algorithm that produces a solution. Goal + preferences

Search is a general problem solving technique for this kind of situations. Search Problem: Input: Initial State + Goal (utility) + actions

(state transitions) Output: A sequence of actions to reach the

goal.

Example: Travel Task

Clean House Task

Vacuum Cleaner Space

Search ProblemSearch Problem

State space Initial state Successor function Goal test Path cost

Example: 8-queensExample: 8-queens

Place 8 queens in a chessboard so that no two queens are in the same row, column, or diagonal.

A solution Not a solution

Example: 8-queensExample: 8-queens

Formulation #1:• States: any arrangement of 0 to 8 queens on the board• Initial state: 0 queens on the board• Successor function: add a queen in any square• Goal test: 8 queens on the board, none attacked

648 states with 8 queens

Example: 8-queensExample: 8-queensFormulation #2:• States: any arrangement of k = 0 to 8 queens in the k leftmost columns with none attacked• Initial state: 0 queens on the board• Successor function: add a queen to any square in the leftmost empty column such that it is not attacked by any other queen• Goal test: 8 queens on the board

2,067 states

Example: Robot Example: Robot navigationnavigation

What is the state space?

Example: Robot Example: Robot navigationnavigation

Cost of one horizontal/vertical step = 1Cost of one diagonal step = 2

Example: Robot Example: Robot navigationnavigation

Example: Robot Example: Robot navigationnavigation

Example: Robot Example: Robot navigationnavigation

Cost of one step = ???

Example: Robot Example: Robot navigationnavigation

Example: Robot Example: Robot navigationnavigation

Example: Robot Example: Robot navigationnavigation

Cost of one step: length of segment

Example: Robot Example: Robot navigationnavigation

Example: Assembly Example: Assembly PlanningPlanning

Initial state

Goal state

Successor function:• Merge two subassemblies

Complex function:it must find if a collision-freemerging motion exists

Example: Assembly Example: Assembly PlanningPlanning

Example: Assembly Example: Assembly PlanningPlanning

Assumptions in Basic Assumptions in Basic SearchSearch

The environment is staticThe environment is discretizableThe environment is observableThe actions are deterministic

open-loop solution

Search Problem Search Problem FormulationFormulation

Real-world environment Abstraction Validity:

Can the solution be executed?

Search Problem Search Problem FormulationFormulation

Real-world environment Abstraction Validity:

Can the solution be executed? Does the state space contain the solution?

Usefulness Is the abstract problem easier than the real-

world problem?

Without abstraction an agent would be swamped by the real world

Search Problem VariantsSearch Problem Variants

One or several initial states One or several goal states The solution is the path or a goal node In the 8-puzzle problem, it is the path to

a goal node In the 8-queen problem, it is a goal node

Any, or the best, or all solutions

Multi-state Search

In vacuum problem, what if we don’t have sensors to tell us whether the floor is clean or not, and which room are we in?

Important ParametersImportant Parameters

Number of states in state space Size of memory needed to store a state Running time of the successor function

ApplicationsApplications

Route finding: airline travel, telephone/computer networks Pipe routing, VLSI routing Pharmaceutical drug design Robot motion planning Video games

SummarySummary

Problem-solving agent State space, successor function, search Examples: Travel Task, house cleaning, 8-queens, robot navigation, assembly planning Assumptions of basic search Important parameters