State-Space Search

CSE 415 -- (c) S. Tanimoto, 2007 Search 1: State Spaces

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State-Space SearchOutline:

Demonstration with T*State spaces, operators, movesA Puzzle: The “Painted Squares”Combinatorics of the PuzzleRepresentations for pieces, boards, and states.Recursive depth-first searchIterative depth-first search


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Introductory Demo

T* = T-STAR = Transparent STate-space search ARchitecture

Missionaries and Cannibals puzzle

Image processing operator sequences

Blocks-World problem (robot motion planning)


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StatesA state consists of a complete description or snapshot of a situation arrived at during the solution of a problem.

Initial state: the starting position or arrangement, prior to any problem-solving actions being taken.

Goal state: the final arrangement of elements or pieces that satisfies the requirements for a solution to the problem.


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State Space

The set of all possible states of the elements or components to be used in solving a problem forms the space of states for the problem. This is known as the state space.


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MovesA move is a transition from one state to another.

An operator is a partial function (from states to states) that can be applied to a state to produce a new state, and also, implicitly, a move.

A sequence of moves that leads from the initial state to a goal state constitutes a solution.


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Operator Preconditions

Precondition: A necessary property of a state in which a particular operator can be applied.

Example: In checkers, a piece may only move into a square that is vacant.Vacant(place) is a precondition on moving a piece into place.

Example: In Chess, a precondition for moving a rook from square A to square B is that all squares between A and B be vacant. (A and B must also be either in the same row or the same column.)


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Painted SquaresHow many possible arrangements?

Could we simply make a list of all the possible arrangements of pieces, and then select those that are solutions?

It depends on how many there are and how much time we can afford to spend.


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The Painted SquaresPuzzle(s)

A Painted Squares Puzzle consists of

1. a board (typically of size 2 by 2), and

2. a set of painted squares. Each square has its sides painted with a texture that is one of: striped, hashed, gray, or boxed.

The objective is to fill the board with the square pieces in such a way that adjacent squares have matching textures where their sides abut one another.


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Factors to consider:1. Number of places on the board2. Number of possible textures in the patterns3. Number of rotations of a square (usually 4)4. Whether flipping squares is allowed5. Symmetries of pieces6. Symmetries of the board7. All possible board shapes for a given side (e.g. the

possible “quadraminos”)8. Full boards only, or partially filled boards, too?


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The number can be reduced by enforcing constraints:

1. Do not permit violations of the matching-sides criterion.

2. Require that filled positions on the board be contiguous and always the first k in some ordering.


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A Solution Approach for the Painted Squares Puzzle

state: partially filled board.

Initial state: empty board.

Goal state: filled board, in which pieces satisfy all constraints.

Operator: for a given remaining piece, given vacancy on the board, and given orientation, try placing that piece at that position in that orientation.


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Recursive Depth-First Method

Current board B empty board.Remaining pieces Q all pieces.Call Solve(B, Q).

Procedure Solve(board B, set of pieces Q)For each piece P in Q, { For each orientation A { Place P in the first available position of B in orientation A, obtaining B’. If B’ is full and meets all constraints, output B’. If B’ is full and does not meet all constraints, return. Call Solve(B’, Q - {P}). }}Return.


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The Combinatorial ExplosionSuppose a search process begins with the initial state.

Then it considers each of k possible moves. Each of those may have k possible subsequent moves.

In order to look n steps ahead, the number of states that must be considered is1 + k + k2 + . . . + kn.

For k > 1, this expression grows rapidly as n increases.(The growth is exponential.) This is known as thecombinatorial explosion.


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Search Trees

By applying operators from a given state we generate its children or successors.

Successors are descendants as are successors of descendants.

If we ignore possible equivalent states among descendants, we get a tree structure.

Depth-First Search: Examine the nodes of the tree by fully exploring the descendants of a node before trying any siblings of a node.


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Graph Search

When descendant nodes can be reached with moves via two or more paths, we are really searching a more general graph than a tree.

Depth-First Search: Examine the nodes of the graph by fully exploring the “descendants” of a node before trying any “siblings” of a node.


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Depth-First Search:Iterative Formulation

1. Put the start state on a list OPEN

2. If OPEN is empty, output “DONE” and stop.

3. Select the first state on OPEN and call it S.

Delete S from OPEN.

Put S on CLOSED.

If S is a goal state, output its description

4. Generate the list L of successors of S and delete

from L those states already appearing on CLOSED.

5. Delete any members of OPEN that occur on L.

Insert all members of L at the front of OPEN.

6. Go to Step 2.


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Breadth-First Search:Iterative Formulation

1. Put the start state on a list OPEN

2. If OPEN is empty, output “DONE” and stop.

3. Select the first state on OPEN and call it S.

Delete S from OPEN.

Put S on CLOSED.

If S is a goal state, output its description

4. Generate the list L of successors of S and delete

from L those states already appearing on CLOSED.

5. Delete any members of OPEN that occur on L.

Insert all members of L at the end of OPEN.

6. Go to Step 2.

State-Space Search

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