CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 1 State-Space Search Outline: Demonstration with T* State spaces, operators, moves A Puzzle: The “Painted Squares” Combinatorics of the Puzzle Representations for pieces, boards, and states. Recursive depth-first search Iterative depth-first search

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 2 Introductory Demo T* = T-STAR = Transparent STate-space search ARchitecture Missionaries and Cannibals puzzle Image processing operator sequences Blocks-World problem (robot motion planning)

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 3 States A 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 4 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 5 Moves A 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 6 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.)

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 7 Painted Squares How 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 8 The Painted Squares Puzzle(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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 9 Painted Squares How many possible arrangements? Factors to consider: 1.Number of places on the board 2.Number of possible textures in the patterns 3.Number of rotations of a square (usually 4) 4.Whether flipping squares is allowed 5.Symmetries of pieces 6.Symmetries of the board 7.All possible board shapes for a given side (e.g. the possible “quadraminos”) 8.Full boards only, or partially filled boards, too?

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 10 Painted Squares How many possible arrangements? 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 11 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 12 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 13 The Combinatorial Explosion Suppose 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 is 1 + k + k k n. For k > 1, this expression grows rapidly as n increases. (The growth is exponential.) This is known as the combinatorial explosion.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 14 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 15 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 16 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.

CSE (c) S. Tanimoto, 2007 Search 1: State Spaces 17 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.