Search in State Spaces• Problem solving as search• Search consists of

– state space– operators– start state– goal states

• A Search Tree is an efficient way to represent a search

• There are a variety of specific search techniques, including– Depth-First, Breadth-First, Best-first search – Others which use heuristic knowledge

Chapter 7 • modeling search

• state space, graph notation, explicit search

Chapter 8• standard search techniques – DFS, BFS.

Chapter 9

• heuristic search• best-first, iterative-deepening, A* algorithm

Chapter 12• adversarial search

• two-player and multiplayer games

What do these problems have in common?• Find the layout of chips on a circuit

board which minimize the total length of interconnecting wires

• Schedule which airplanes and crew fly to which cities for American, United, British Airways, etc

• Write a program which can play chess against a human

•Build a system which can find human faces in an arbitrary digital image

• Program a tablet-driven portable computer to recognize your handwriting

• Decrypt data which has been encrypted but you do not have the key

• Answer–they can all be formulated as search problems

Search in State Spaces• Many problems in Artificial Intelligence can be

mapped onto searches in particular state spaces.

• This concept is especially useful if the system (our “world”) can be defined as having a finite number of states, including an initial state and one or more goal states.

• Optimally, there are a finite number of actions that we can take, and there are well-defined state transitions that only depend on our current state and current action.

Search in State Spaces• Let us consider an easy task in a very simple world with our robot being the only actor in it:• The world contains a floor and three toy blocks labeled A, B, and C.• The robot can move a block (with no other block on top of it) onto the floor or on top of another block.• These actions are modeled by instances of a schema, move(x, y).• Instances of the schema are called operators.

Search in State Spaces• The robot’s task is to stack the toy blocks so that A

is on top of B, B is on top of C, and C is on the floor.• For us it is clear what steps have to be taken to

solve the task.• The robot has to use its world model to find a

solution.• Let us take a look at the effects that the robot’s

actions exert on its world.

Setting Up a State Space Model• State-space Model is a Model for

The Search Problem– usually a finite set of states X

•e.g., in driving, the states in the model could be towns/cities

•in games, board positions

• Start State - a state from X where the search starts

• Goal State(s) a goal is defined as a target state

– For now: all goal states have utility 1, and all non-goals have utility 0– there may be many states which satisfy the goal

• e.g., drive to a town with an airport– or just one state which satisfies the goal

• e.g., drive to Las Vegas

• Operators– operators are mappings from X to X

• e.g. moves from one city to another that are legal (connected by a road)

Example

There are three blocks named A, B and C on a table. You can take any (one) block and move it on top of any other stack or to the table. Goal is to get the stack A, B, C.

Start:

Final:

Search in State Spaces

•Effects of moving a block (illustration and list-structure iconic model notation)

Search in State Spaces• In order to solve the task efficiently, the robot

should “look ahead”, that is simulate possible actions and their outcomes.

• Then, the robot can carry out a sequence of actions that, according to the robot’s prediction, solves the problem.

• A useful structure for such a simulation of alternative sequences of action is a directed graph.

• Such a graph is called a state-space graph.

State-Space Graphs

State-Space Graphs• To solve a particular problem, the robot has to find a

path in the graph from a start node (representing the initial state) to a goal node (representing a goal state).

• The resulting path indicates a sequence of actions that solves the problem.

• The sequence of operators along a path to a goal is called a plan.

• Searching for such a sequence is called planning.

• Predicting a sequence of world states from a sequence of actions is called projecting.

State-Space Graphs• There are various methods for searching state

spaces.• One possibility is breadth-first search:

– Mark the start node with a 0.– Mark all adjacent nodes with 1.– Mark all unmarked nodes adjacent to

nodes with a 1 with the number 2, and so on, until you arrive at a goal node.

– Finally, trace a path back from the goal to the start along a sequence of decreasing numbers.

Decision Trees•A decision tree is a special case of a state-space graph.•It is a rooted tree in which each internal node corresponds to a decision, with a subtree at these nodes for each possible outcome of the decision.•Decision trees can be used to model problems in which a series of decisions leads to a solution. •The possible solutions of the problem correspond to the paths from the root to the leaves of the decision tree.

Example: There are 12 coins of which one is heavier or lighter than the rest. Find the odd coin using a balance with 3 weighings.

Decision Trees• Example: The n-queens problem•How can we place n queens on an nn chessboard so that no two queens can capture each other?

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A queen can move any A queen can move any number of squares number of squares horizontally, vertically, and horizontally, vertically, and diagonally.diagonally.

Here, the possible target Here, the possible target squares of the queen Q are squares of the queen Q are marked with an x.marked with an x.

Decision Trees•Let us consider the 4-queens problem.• Question: How many possible configurations of 44 chessboards containing 4 queens are there?•Answer: There are 16!/(12!4!) = (13141516)/(234) = 13754 = 1820 possible configurations.• Shall we simply try them out one by one until we encounter a solution?•No, it is generally useful to think about a search problem more carefully and discover constraints on the problem’s solutions.•Such constraints can dramatically reduce the size of the relevant state space.

Decision Trees• Obviously, in any solution of the n-queens problem, there must be exactly one queen in each column of the board. • Otherwise, the two queens in the same column could capture each other.• Therefore, we can describe the solution of this problem as a sequence of n decisions: • Decision 1: Place a queen in the first column.• Decision 2: Place a queen in the second column... Decision n: Place a queen in the n-th column.•

Backtracking in Decision Trees

• There are problems that require us to perform an exhaustive search of all possible sequences of decisions in order to find the solution.

• We can solve such problems by constructing the complete decision tree and then find a path from its root to a leave that corresponds to a solution of the problem (breadth-first search often requires the construction of an almost complete decision tree).

• In many cases, the efficiency of this procedure can be dramatically increased by a technique called backtracking (depth-first search).

Backtracking in Decision Trees

•Idea: Start at the root of the decision tree and move downwards, that is, make a sequence of decisions, until you either reach a solution or you enter a situation from where no solution can be reached by any further sequence of decisions.•In the latter case, backtrack to the parent of the current node and take a different path downwards from there. If all paths from this node have already been explored, backtrack to its parent.•Continue this procedure until you find a solution or establish that no solution exists (there are no more paths to try out).

Backtracking in Decision Trees

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place 1place 1stst queen queen

place 2place 2ndnd queen queen

place 3place 3rdrd queen queen

place 4place 4thth queen queen

empty boardempty board

Summary: Defining Search Problems

• A statement of a Search problem has 4 components– 1. A set of states– 2. A set of “operators” which allow one to get

from one state to another– 3. A start state S– 4. A set of possible goal states, or ways to test

for goal states

• Search solution consists of– a sequence of operators which transform S into

a a unique goal state G. (often, we may want the shortest path to the goal.)