Search Problems: Concepts and Examples

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the primary function of search in reasoning as described?

To find a definitive solution immediately.

To explore alternatives. (correct)

To eliminate options without evaluation.

To create a static state.

In the example of the 8-Puzzle, what does the initial state represent?

A state that has reached the end goal.

The first move made by the player.

The position of the tiles in the goal configuration.

Any arrangement of 8 numbered tiles and an empty tile. (correct)

What does the successor function in the 8-Puzzle provide?

A guide to determine the best move to make next.

The definitive final state of the game.

Knowledge about the possible subsequent states. (correct)

A method to track the player's previous moves.

How does reasoning relate to declarative knowledge in the context of robotics?

Reasoning utilizes declarative knowledge to create alternatives. Signup and view all the answers

What is NOT a characteristic of search in the context of the provided content?

It eliminates the need for alternatives. Signup and view all the answers

What does the actual state space consist of?

The set of all states. Signup and view all the answers

Why is it often not feasible to build a complete representation of the state graph?

The number of states can be astronomically large. Signup and view all the answers

How many states does the 15-puzzle contain?

2.09 x 10^13 states. Signup and view all the answers

Which is the fastest test for determining if one state is reachable from another?

Applying a fast test that focuses on a small portion of the graph. Signup and view all the answers

What is a significant drawback of traditional search techniques?

They can be inefficient when there is no solution. Signup and view all the answers

What is the estimated time taken to explore the entirety of the 15-puzzle state space?

55 hours. Signup and view all the answers

When solving state space problems, what must a problem solver typically do?

Construct a solution by exploring just a small portion of the graph. Signup and view all the answers

What is the state space of the 24-puzzle estimated to be?

10^25 states. Signup and view all the answers

What is the state space size of the 8-puzzle?

362,880 states Signup and view all the answers

How many states are reachable from a particular state in the 15-puzzle?

Only half of the states are reachable Signup and view all the answers

What does the variable N represent in the context of the (n^2-1)-puzzle?

The sum of permutation inversions and row number of the empty tile Signup and view all the answers

What invariant property does (N mod 2) maintain during legal moves of the empty tile?

It remains unchanged Signup and view all the answers

For a goal state to be reachable from a current state in the (n^2-1)-puzzle, what must be true about N?

N(g) and N(s) must have the same parity Signup and view all the answers

What is the approximate state space size of the 15-puzzle?

2.09 x 10^13 states Signup and view all the answers

Which of the following statements about the empty tile in the (n^2-1)-puzzle is true?

The empty tile affects the parity of N Signup and view all the answers

How many connected components exist in the state graph of the (n^2-1)-puzzle?

Two components of equal size Signup and view all the answers

What does the state space represent?

An abstract representation of collection of possible worlds sharing crucial properties. Signup and view all the answers

Which statement about the successor function is accurate?

It only returns the successor states and their respective costs. Signup and view all the answers

What is the role of path cost in a problem-solving process?

To measure the cost of performing the action corresponding to an arc. Signup and view all the answers

Under what condition do we accept the cost of any arc in a problem?

The cost must be a positive number and greater than a constant epsilon. Signup and view all the answers

In assembly planning, how is the successor function characterized?

As a complex function involving collision and stability considerations. Signup and view all the answers

Why is it significant that the cost of an arc must always be greater than epsilon (ε)?

To guarantee that longer paths incur higher costs. Signup and view all the answers

What is indicated by the term 'black box' regarding the successor function?

Its internal workings are unknown and not distinguished. Signup and view all the answers

What characterizes a solution to a search problem?

It involves a path with minimum cost from the initial node to a goal node. Signup and view all the answers

What does assuming that the cost c of an arc is always c ≥ ε > 0 prevent?

It prevents the cost from being arbitrarily small. Signup and view all the answers

In the context of search problems, what is the significance of the state space S?

It defines all possible states and transitions in the search process. Signup and view all the answers

Which of the following games originated in China over 3000 years ago?

Go Signup and view all the answers

What is true about the arc cost in a search problem?

It can vary and contributes to the total cost of a path. Signup and view all the answers

Who introduced the 15-Puzzle and claimed to be America's greatest puzzle-expert?

Sam Loyd Signup and view all the answers

What does the goal test determine in a search problem?

If a state is the goal or not. Signup and view all the answers

What was unique about Sam Loyd's offer regarding the 15-Puzzle?

No one ever won the prize he offered. Signup and view all the answers

What does a state graph in a search problem consist of?

Nodes representing different potential states connected by arcs. Signup and view all the answers

What is the cost of one diagonal step according to the formulations?

√2 Signup and view all the answers

What is the goal state in the assembly (sequence) planning formulation?

Un-decomposed assembly Signup and view all the answers

Which assumption is NOT a part of basic search?

The actions are random Signup and view all the answers

What does the arc cost in the assembly planning refer to?

Time taken to carry out a merging Signup and view all the answers

Which feature of the state space is ruled out by the formulation in assembly planning?

Non-monotonic assemblies Signup and view all the answers

What is required for a successor of a state in assembly planning?

Checking if the merging is feasible Signup and view all the answers

In formulation #3, how is the cost of a step defined?

Length of the segment Signup and view all the answers

What is usually needed after obtaining a solution path in the context provided?

Path-smoothing post-processing Signup and view all the answers

Which of the following assumptions can be removed while still allowing search to be effective?

World is static Signup and view all the answers

What is indicated by the term 'visibility graph' in the provided content?

A model connecting visible regions for navigation Signup and view all the answers

Study Notes

Search Problems

Search problems involve exploring alternative solutions.
Reasoning in these problems involves exploring choices.
R&N: Chap. 3, Sect. 3.1-2 + 3.6

Components of Search Problems

Declarative knowledge provides potential options.
Which pieces of knowledge are useful?
How to use the knowledge?
Search focuses on exploring alternative options.
It is a vital approach to using knowledge effectively.

Example: 8-Puzzle

State: An arrangement of numbered tiles (1-8) and an empty space on a 3x3 board.
Initial state: A specific starting arrangement of tiles.
Goal state: A desired arrangement of tiles.

8-Puzzle: Successor Function

SUCC(state): A set of possible next states from a given state.
The successor function describes how to generate possible next steps from a given state.
This does not specify which move to make from the current state.

Historical Examples of Search Problems

Chess: Originated in Persia and India, approximately 4000 years ago.
Checkers: Found in 3600-year-old Egyptian paintings.
Go: Originated in China, over 3000 years ago.
Puzzles and games requiring exploration of alternatives are a crucial part of human intelligence.

(n² - 1)-Puzzle

Generalization of the 8-puzzle.
Examples include 15-puzzle, 24-puzzle and so on
Large state spaces.

15-Puzzle

Introduced in 1878 by Sam Loyd, claiming to be America's greatest puzzle expert
A $1,000 prize was offered; no one ever won this.
15 numbered tiles and an empty space on a 4x4 board.

Stating a Problem as a Search Problem

State space (S): The set of all possible states in the problem.
Successor function: A function that, given a state, returns a set of possible next states.
Initial state (s₀): The starting state of the problem.
Goal test: A function to check if a given state is a goal state.
Arc cost: A measure of the cost to move from one state to another.

State Graph

Each state is represented as a node.
An arc (or edge) connects two nodes if the second node is a successor of the first.
State graphs can comprise multiple disconnected components.

Solution to the Search Problem

A solution is a path from the initial node to a goal node.
The cost of a path is the sum of arc costs along that path.
An optimal solution is a solution path with minimum cost.

Size of State Space

8-puzzle: 362,880 possible states.
15-puzzle: 2.09 x 10¹³ possible states.
24-puzzle: 10²⁵ possible states.
Only half of the states are reachable in the 15-puzzle, and 24-puzzle.

Permutation Inversions

Number of inversions (N) is used to determine the parity of a state.

Proposition: (N mod 2)

Parity (N mod 2) remains unchanged with any legal move of the empty tile.
This is vital for determining reachability.

Applications of Search

Route finding (airline travel, networks)
Package/mail delivery
Pipe routing
Protein fold comparisons
Pharmaceutical drug design
Video games

Simple Problem-Solving Agent Algorithm

Steps to solve a problem through search.
- I = intial state, GOAL? = goal test, Succ = successor function.
- solution = search(I, GOAL?, Succ)
- Perform the solution.

Assumptions in Search

Static world.
Discretizable world.
Observable world.
Deterministic actions.

Path Planning

Example of problem formulated using different methods (state spaces and successor functions) in different contexts.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers search problems, focusing on alternative solutions and decision-making processes. Key concepts include declarative knowledge, the 8-puzzle problem, and the successor function. Explore the significance of exploring options and effective knowledge usage.